DC Algorithm Research Articles

A difference of convex functions algorithm for optimal scheduling and real-time assignment of preventive maintenance jobs on parallel processors

In this paper, we introduce a new approach based on DC (Difference of Convexfunctions) Programming and DCA (DC Algorithm) for minimizing the maintenancecost involving flow-time and tardiness penalties by optimal scheduling and real-timeassignment of preventive maintenance jobs on parallel processors. The main idea is to dividethe horizon considered into $H$ intervals. The problem is first formulatedas a mixed integer linear problem (MILP) and then reformulated as a DCprogram. A solution method based on DCA is used to solve the resultingproblem. The efficiency of DCA is compared with the algorithm based on thenew flow-time and tardiness rule (FTR) given in [1]. Thecomputational results on several test problems show that the solutionsprovided by DCA are better.

DCA based algorithms for multiple sequence alignment (MSA)

In the last years many techniques in bioinformatics have been developed for the central and complex problem of optimally aligning biological sequences. In this paper we propose a new optimization approach based on DC (Difference of Convex functions) programming and DC Algorithm (DCA) for the multiple sequence alignment in its equivalent binary linear program, called “Maximum Weight Trace” problem. This problem is beforehand recast as a polyhedral DC program with the help of exact penalty techniques in DC programming. Our customized DCA, requiring solution of a few linear programs, is original because it converges after finitely many iterations to a binary solution while it works in a continuous domain. To scale-up large-scale (MSA), a constraint generation technique is introduced in DCA. Preliminary computational experiments on benchmark data show the efficiency of the proposed algorithm DCAMSA, which generally outperforms some standard algorithms.

New and efficient DCA based algorithms for minimum sum-of-squares clustering

The purpose of this paper is to develop new efficient approaches based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) to perform clustering via minimum sum-of-squares Euclidean distance. We consider the two most widely used models for the so-called Minimum Sum-of-Squares Clustering (MSSC in short) that are a bilevel programming problem and a mixed integer program. Firstly, the mixed integer formulation of MSSC is carefully studied and is reformulated as a continuous optimization problem via a new result on exact penalty technique in DC programming. DCA is then investigated to the resulting problem. Secondly, we introduce a Gaussian kernel version of the bilevel programming formulation of MSSC, named GKMSSC. The GKMSSC problem is formulated as a DC program for which a simple and efficient DCA scheme is developed. A regularization technique is investigated for exploiting the nice effect of DC decomposition and a simple procedure for finding good starting points of DCA is developed. The proposed DCA schemes are original and very inexpensive because they amount to computing, at each iteration, the projection of points onto a simplex and/or onto a ball, and/or onto a box, which are all determined in the explicit form. Numerical results on real word datasets show the efficiency, the scalability of DCA and its great superiority with respect to k-means and kernel k-means, standard methods for clustering.

New and efficient algorithms for transfer prices and inventory holding policies in two-enterprise supply chains

We consider a multi-period problem of fair transfer prices and inventory holding policies in two enterprise supply chains. This problem was formulated as a mixed integer non-linear program by Gjerdrum et al. (Eur J Oper Res 143:582---599, 2002). Existing global optimization methods to solve this problem are computationally expensive. We propose a continuous approach based on difference of convex functions (DC) programming and DC Algorithms (DCA) for solving this combinatorial optimization problem. The problem is first reformulated as a DC program via an exact penalty technique. Afterward, DCA, an efficient local approach in non-convex programming framework, is investigated to solve the resulting problem. For globally solving this problem, we investigate a combined DCA-Branch and Bound algorithm. DCA is applied to get lower bounds while upper bounds are computed from a relaxation problem. The numerical results on several test problems show that the proposed algorithms are efficient: DCA provides a good integer solution in a short CPU time although it works on a continuous domain, and the combined DCA-Branch and Bound finds an $$\epsilon $$ ∈ -optimal solution for large-scale problems in a reasonable time.

Efficient DC programming approaches for the asymmetric eigenvalue complementarity problem

In this paper, we propose nonlinear programming (NLP) formulations and difference of convex functions (DC) programming approaches for the asymmetric eigenvalue complementarity problem (EiCP). The EiCP has a solution if and only if these NLP formulations have zero global optimal value. We reformulate the NLP formulations as DC programs which can be efficiently solved by a DC algorithm. Some preliminary numerical results illustrate the good performance of the proposed methods.

Block Clustering Based on Difference of Convex Functions (DC) Programming and DC Algorithms

We investigate difference of convex functions (DC) programming and the DC algorithm (DCA) to solve the block clustering problem in the continuous framework, which traditionally requires solving a hard combinatorial optimization problem. DC reformulation techniques and exact penalty in DC programming are developed to build an appropriate equivalent DC program of the block clustering problem. They lead to an elegant and explicit DCA scheme for the resulting DC program. Computational experiments show the robustness and efficiency of the proposed algorithm and its superiority over standard algorithms such as two-mode K-means, two-mode fuzzy clustering, and block classification EM.

High-Efficiency Isolated Bidirectional AC–DC Converter for a DC Distribution System

A high-efficiency isolated bidirectional ac–dc converter is proposed for a 380-V dc power distribution system to control bidirectional power flows and to improve its power conversion efficiency. To reduce the switches’ losses of the proposed nonisolated full-bridge ac–dc rectifier using an unipolar switching method, switching devices employ insulated-gate bipolar transistors, MOSFETs, and silicon carbide diodes. Using the analysis of the rectifier’s operating modes, each switching device can be selected by considering switch stresses. A simple and intuitive frequency detection method for a single-phase synchronous reference frame-phase-locked loop (SRF-PLL) is also proposed using a filter compensator, a fast period detector, and a finite impulse response filter to improve the robustness and accuracy of PLL performance under fundamental frequency variations. In addition, design and control methodology of the bidirectional full-bridge CLLC resonant converter is suggested for the galvanic isolation of the dc distribution system. A dead-band control algorithm for the bidirectional dc–dc converter is developed to smoothly change power conversion directions only using output voltage information. Experimental results will verify the performance of the proposed methods using a 5-kW prototype converter.

Long-Short Portfolio Optimization Under Cardinality Constraints by Difference of Convex Functions Algorithm

In the matter of Portfolio selection, we consider an extended version of the Mean-Absolute Deviation (MAD) model, which includes discrete asset choice constraints (threshold and cardinality constraints) and one is allowed to sell assets short if it leads to a better risk-return tradeoff. Cardinality constraints limit the number of assets in the optimal portfolio and threshold constraints limit the amount of capital to be invested in (or sold short from) each asset and prevent very small investments in (or short selling from) any asset. The problem is formulated as a mixed 0---1 programming problem, which is known to be NP-hard. Attempting to use DC (Difference of Convex functions) programming and DCA (DC Algorithms), an efficient approach in non-convex programming framework, we reformulate the problem in terms of a DC program, and investigate a DCA scheme to solve it. Some computational results carried out on benchmark data sets show that DCA has a better performance in comparison to the standard solver IBM CPLEX.

Solving continuous min max problem for single period portfolio selection with discrete constraints by DCA

In this article, we consider the application of a continuous min–max model with cardinality constraints to worst-case portfolio selection with multiple scenarios of risk, where the return forecast of each asset belongs to an interval. The problem can be formulated as minimizing a convex function under mixed integer variables with additional complementarity constraints. We first prove that the complementarity constraints can be eliminated and then use Difference of Convex functions (DC) programming and DC Algorithm (DCA), an innovative approach in non-convex programming frameworks, to solve the resulting problem. We reformulate it as a DC program and then show how to apply DCA to solve it. Numerical experiments on several test problems are reported that demonstrate the accuracy of the proposed algorithm.

On an outer approximation algorithm for the canonical DC progmamming problem.

On an outer approximation algorithm for the canonical DC progmamming problem.

A DC programming approach for planning a multisensor multizone search for a target

In this paper, we consider a well-known problem in the general area of search theory: planning a multisensor in multizone search so as to maximize the probability of detection of a target under a given resource effort to be shared. We propose a new optimization model that is a nonlinear mixed 0–1 programming problem. This problem is then reformulated as a DC (Difference of Convex) functions program via an exact penalty technique. DC programming and DCA (DC algorithm) have been investigated for solving the resulting DC program. Numerical experiments demonstrate the efficiency and the superiority of the proposed algorithm in comparison with the existing method.

Joint Optimization of Source Power Allocation and Distributed Relay Beamforming in Multiuser Peer-to-Peer Relay Networks

In this paper, we consider the joint optimization of the source power allocation and relay beamforming weights in distributed multiuser peer-to-peer (MUP2P) relay networks applying the amplify-and-forward (AF) protocol. We adopt a quality-of-service (QoS) based approach, in which the total power transmitted from all sources and relays is minimized while guaranteeing the prescribed QoS requirement of each source-destination pair. The QoS is modeled as a function of the receive signal-to-interference-plus-noise ratio (SINR) at the destinations. Unlike the existing contributions, the transmitted powers of the sources and the beamforming weights of the relays are optimized jointly in this paper. Introducing an appropriate transformation of variables, the QoS based source power allocation and distributed relay beamforming (PADB) problem can be equivalently transformed into a difference of convex (DC) program, which can be efficiently solved with local optimality using the constrained concave convex procedure (CCCP). Based on this procedure, we also propose an iterative feasibility search algorithm (IFSA) to find an initial feasible point of the DC program. The analytic study of the proposed solution confirms that it converges to a local optimum of the PADB problem. Numerical results show that our solution outperforms (in terms of the total transmitted power) the alternating optimization procedure and the exact penalty based DC algorithm. In addition, the proposed IFSA outperforms the alternating optimization algorithm in finding feasible points of the DC program (i.e., the equivalence of the PADB problem).

A Proof of Convergence of the Concave-Convex Procedure Using Zangwill's Theory

The concave-convex procedure (CCCP) is an iterative algorithm that solves d.c. (difference of convex functions) programs as a sequence of convex programs. In machine learning, CCCP is extensively used in many learning algorithms, including sparse support vector machines (SVMs), transductive SVMs, and sparse principal component analysis. Though CCCP is widely used in many applications, its convergence behavior has not gotten a lot of specific attention. Yuille and Rangarajan analyzed its convergence in their original paper; however, we believe the analysis is not complete. The convergence of CCCP can be derived from the convergence of the d.c. algorithm (DCA), proposed in the global optimization literature to solve general d.c. programs, whose proof relies on d.c. duality. In this note, we follow a different reasoning and show how Zangwill's global convergence theory of iterative algorithms provides a natural framework to prove the convergence of CCCP. This underlines Zangwill's theory as a powerful and general framework to deal with the convergence issues of iterative algorithms, after also being used to prove the convergence of algorithms like expectation-maximization and generalized alternating minimization. In this note, we provide a rigorous analysis of the convergence of CCCP by addressing two questions: When does CCCP find a local minimum or a stationary point of the d.c. program under consideration? and when does the sequence generated by CCCP converge? We also present an open problem on the issue of local convergence of CCCP.

Binary classification via spherical separator by DC programming and DCA

In this paper, we consider a binary supervised classification problem, called spherical separation, that consists of finding, in the input space or in the feature space, a minimal volume sphere separating the set $${\mathcal{A}}$$ from the set $${\mathcal{B}}$$ (i.e. a sphere enclosing all points of $${ \mathcal{A}}$$ and no points of $${\mathcal{B}}$$ ). The problem can be cast into the DC (Difference of Convex functions) programming framework and solved by DCA (DC Algorithm) as shown in the works of Astorino et al. (J Glob Optim 48(4):657---669, 2010). The aim of this paper is to investigate more attractive DCA based algorithms for this problem. We consider a new optimization model and propose two interesting DCA schemes. In the first scheme we have to solve a quadratic program at each iteration, while in the second one all calculations are explicit. Numerical simulations show the efficiency of our customized DCA with respect to the methods developed in Astorino et al.

Forecasting foreign exchange rates using kernel methods

First, the all-important no free lunch theorems are introduced. Next, kernel methods, support vector machines (SVMs), preprocessing, model selection, feature selection, SVM software and the Fisher kernel are introduced and discussed. A hidden Markov model is trained on foreign exchange data to derive a Fisher kernel for an SVM, the DC algorithm and the Bayes point machine (BPM) are also used to learn the kernel on foreign exchange data. Further, the DC algorithm was used to learn the parameters of the hidden Markov model in the Fisher kernel, creating a hybrid algorithm. The mean net returns were positive for BPM; and BPM, the Fisher kernel, the DC algorithm and the hybrid algorithm were all improvements over a standard SVM in terms of both gross returns and net returns, but none achieved net returns as high as the genetic programming approach employed by Neely, Weller, and Dittmar (1997) and published in Neely, Weller, and Ulrich (2009). Two implementations of SVMs for Windows with semi-automated parameter selection are built.

A Branch‐and‐Bound Algorithm Embedded with DCA for DC Programming

The special importance of Difference of Convex (DC) functions programming has been recognized in recent studies on nonconvex optimization problems. In this work, a class of DC programming derived from the portfolio selection problems is studied. The most popular method applied to solve the problem is the Branch‐and‐Bound (B&B) algorithm. However, “the curse of dimensionality” will affect the performance of the B&B algorithm. DC Algorithm (DCA) is an efficient method to get a local optimal solution. It has been applied to many practical problems, especially for large‐scale problems. A B&B‐DCA algorithm is proposed by embedding DCA into the B&B algorithms, the new algorithm improves the computational performance and obtains a global optimal solution. Computational results show that the proposed B&B‐DCA algorithm has the superiority of the branch number and computational time than general B&B. The nice features of DCA (inexpensiveness, reliability, robustness, globality of computed solutions, etc.) provide crucial support to the combined B&B‐DCA for accelerating the convergence of B&B.

Dynamic Spectrum Management (DSM) Algorithms for Multi-User xDSL

Dynamic Spectrum Management (DSM) is an effective method for reducing the effect of crosstalk in Digital Subscriber Line (DSL) systems. This paper discusses various DSM algorithms, including Optimal Spectrum Balancing (OSB), Iterative Spectrum Balancing (ISB), Autonomous Spectrum Balancing (ASB), Iterative Water-Filling (IWF), Selective Iterative Water-filling (SIW), Successive Convex Approximation for Low complExity (SCALE), the Difference of Convex functions Algorithm (DCA), and Distributed Spectrum Balancing (DSB). They are compared in terms of performance (achievable data rate) and computational complexity.

Power flow management algorithm for photovoltaic systems feeding DC/AC loads

In this paper, a power flow management algorithm for photovoltaic system feeding DC and AC loads is presented. PV being a major energy source enables the DC loads to be connected directly to the DC bus. A grid connected PV system involves one power source (PV array), one power sink (load) and two power sources/sink (utility and battery), and hence a power flow management system is required to balance the power flow among these sources. A unique control algorithm is developed for selecting the operating mode of the bidirectional converter by sensing the battery voltage. The bidirectional converter has three modes of operation namely, the rectifier mode, inverter mode and the voltage control mode. In both the rectifier and inverter modes of operation, the converter current is controlled by employing a hysteresis current controller and unity power factor is achieved in both the cases. In voltage control mode, the converter regulates the inverter voltage and supplies the reactive power to the grid. Detailed simulation studies are carried out on a single phase bidirectional converter in MATLAB/Simulink environment. The viability of the scheme has been ascertained by performing experimental studies on a laboratory prototype. The control strategy is digitally implemented on an Altera Cyclone II FPGA board and the algorithm is verified for different modes of operation by varying the load and the relevant results are presented.

Solving Partitioning-Hub Location-Routing Problem using DCA

The Partitioning-Hub Location-Routing Problem (PHLRP) is a hub location problem involving graph partitioning and routing features. PHLRP consists of partitioning a given network into sub-networks, locating at least one hub in each sub-network and routing the traffic within the network at minimum cost. There are various important applications of PHLRP, such as in the deployment of network routing protocol problems and in the planning of freight distribution problems. We first present the formulation of this problem as an Binary Integer Linear Programming (BILP) and then investigate a new method based on DC (Difference of Convex functions) programming and DCA (DC Algorithms). Preliminary numerical results are compared with CPLEX, the best solver for BILP. These results show that the proposed algorithm is efficient.

Efficacy and Safety Testing of a New Biologically Based Design Ankle Foot Orthosis in Healthy Volunteer

The ankle-foot of human body is a multi-joint structure that accommodates complex foot motion. Abnormality to the ankle-foot due to injury or disease can result in abnormal gait motion. In such cases, physiotherapist has to assist hemiplegic patients (ankle dorsiflexor muscles with lack of dorsiflexion assist moment) in rehabilitation therapy by using gait training in parallel bars. Physiotherapist has to support hemiplegic patient to position foot and also supports their stand balance. This prolongs multiple task puts extra burden to physiotherapist which gives side effect such as muscular strain or bone fracture while doing the task. Consequently, the motion of the foot patients did not follow the normal gait pattern. Therefore, there is a need to develop an effective ankle foot orthosis (AFO) to solve the long issue-problem. This research was undertake to embark on the modeling and designing of new ankle foot orthosis (AFO) using active control system which later could be used to help patients with ankle dorsiflexor muscles problem. The work was carried out in four stages involving modeling and simulation of DC motor, algorithm development, design and fabrication of the orthosis and finally, evalaution of the product and its functions. The orthosis was tested on healthy volunteer and the results show that the objective to develop and fabricate a new type of robust ankle foot orthosis which can control movement has been achieved successfully.